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Looking at the Pazzi Chapel's Umbrella Vault from Its Oculus

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Looking at the Pazzi Chapel’s umbrella vault from its oculus Nevena RADOJEVIC DiDA: Dipartimento di Architettura – Università degli Studi di Firenze, Italy Abstract: The subject of the research is the umbrella vault in the Pazzi Chapel in Santa Croce church in Florence. In some previous researches done by the author, attempts were made to formulate new hypotheses on the form-finding processes that could have determined the shape of umbrella’s vault sail. The analyses were done by correlating detailed surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form. The hypothesized form of the inner sail is obtained by a three-dimensional transformation of a toric surface, with respect to the fixed point (dome’s oculus) and constant length (torus radius), like the conchoid of Nicomedes in 2d. The vault is made by the inner sail (the conchoid surface) and the outer sail that is not visible in this moment (the hypothesized toric surface). The results were verified by overlapping the hypothesized inner sail’s form with the laser scanner data, and only 4% of points were out of the 3 cm range. This very particular form, even though it explains the building process in a very satisfactory manner, is not actually something that the visitors can see (with the naked eye). Actually, the conchoid surface (that is a solid central projection of torus), could be perceived like a transformation of torus only under certain conditions. First condition: the inner conchoidal sail’s surface should be mapped by the elements that represent the projection of the torus’s peculiar elements (circumferences for example) onto the conchoid surface from the fixed point (oculus). These irregular spatial curves, that represent the regular curve’s transformation, can be perceived as if they were circular if we look at them from the transformation center. The second condition would be the transportation of the observer’s view point in the center of oculus (not accessible point). The exhibition design (work in progress for the Opera of Santa Croce) involves the use of projectors for the 3d mapping of the sails and use of a plane mirrors system that can move the observer’s view point as if he was in the cupola’s oculus. Keywords: 3d Conchoid surface, Torus, Mirror, 3d Mapping, Solid central projection Introduction The Pazzi Chapel in Santa Croce church in Florence is considered one of the cornerstones of Renaissance architecture, and to most scholars, it marks one of the highlights of the career of the great architect Filippo Brunelleschi, who is best known as the man who designed the dome of the cathedral in Florence. The chapel, put up in the mid-1400's by the Medici rivals, the Pazzi banking family, seems to sum up his ideas of Renaissance architecture perfectly. The building was started between 1429 and 1430, but the date in which it was completed is not certain and it ranges from 1443 and 1478, in any case after Brunelleschi’s death (1446). The structural system is the same like one he adopted in the Old Sacristy of San Lorenzo and Duomo’s cupola afterwards, and it’s based on a double-sailed vault structure. International Conference on Cultural Heritage and New Technologies | Vienna | 2015 According to some scholars it was not Brunelleschi that designed the Pazzi Chapel but Michelozzo that copied it from the Old Sacristy: Marvin Trachtenberg argued that is unlikely that an architect as powerful and influential as Brunelleschi would have allowed an immature design and it is all but unthinkable that so potent and creative force as Brunelleschi would actually have offered up an unrefined copy of his early work (TRACHTENBERG 1996, 1997, 2008). In the Pazzi Chapel, Trachtenberg points out, there is little of the direct and visible connection between ornament and structure that helps make the Old Sacristy so notable; and the ornament, while superficially similar to that of the Old Sacristy, has ''a surprising roughness and clumsiness'' (TRACHTENBERG 1997). The Trachtenberg’s arguments are based on evidences that came through an intense study of the building itself, especially its ornaments and structure. Fig. 1 – Pazzi Chapel in Santa Croce Church in Florence: external view from the cloister (above) and internal view of umbrella vault (below). For years, scholars have offered theories about why the chapel was not entirely up to Brunelleschi's usual level of creative genius, but these tended to be somewhat contrived attempts to explain the master's lapse . However, no one of these studies considered the geometrical characteristics of the vaults, based on detailed surveys. In some previous researches done by author, attempts were made to formulate new hypotheses on the form- finding processes that could have determined the shape of umbrella’s vault sail. The analyses were done by correlating detailed digital surveys 1 and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form. For the complexity of argument that was treated in some previous papers (RADOJEVIC 2014, 2015), in this occasion we will give just a brief explanation of the design processes that led to the actual form, necessary to understand for the comprehension of this contribution. ______ 1 The integrated survey (terrestrial laser scanner and photogrammetry), promoted by Opera di Santa Crocec was done in 2012 in a collaboration with Università di Firenze within the research project “Laboratorio di Santa Croce” . The survey team was composed by prof. Maria Teresa Bartoli and the author. 2 Radojevic – Looking at the Pazzi Chapel’s umbrella vault from its oculus Geometry of the umbrella vault The umbrella vault’s is done on dodecagonal base, and the semi-circular stone ribs support the light double- sailed structure. The ribs are radiating from center of cupola where there is a circular lantern. The sails are double curved with a very complex geometry. Currently, we can see (and measure) only the inner sail’s intrados; but we have some testimonials about the outer one. The survey designs, done by P. A. Rossi during the roof restauration period, show the whole structure, (fig. 3). From these documents we can see that the umbrella vault is double sailed and that the ribs are done by the radial brick layering (LASCHI, ROSSELLI and ROSSI 1962). The hypothesis made by the author considers the form of the inner sail that is obtained by a three-dimensional transformation of a toric surface with respect to the fixed point (dome’s oculus) and constant length (torus radius); like the conchoid of Nicomedes in 2d. Fig. 2 – Point cloud snapshots that show the top orthographic view and the longitudinal section of Pazzi Chapel (on the left) and the three cupolas in this order (from top to bottom): the portico vault (related to the stereographic projection), the umbrella vault (torus solid projection) and the semispherical dome with astronomical fresco. 3 International Conference on Cultural Heritage and New Technologies | Vienna | 2015 Fig. 3 – The sequence of images show the survey drawings done by arch. Paolo Alberto Rossi during the restauration works. From these documents we can see that the umbrella vault is double sailed and that the ribs are done by the radial brick layering. (Drawings by: P. A. Rossi) a) Structural perspective drawing of the umbrella vault that shows the double sailed structure. b) Axonometric drawing that shows a rib’s structure. c) Analytic structural diagram of the rib. (LASCHI, ROSSELLI and ROSSI 1962) According to this hypothesis, the vault is made by the inner sail (the conchoid surface) and the outer sail that is currently not visible (the hypothesized toric surface). Thus obtained nurbs surface was overlapped by the mesh surface generated from the surveying data (the cloud of points). The models overlapped almost perfectly! Only 4% of points from the mesh model were out of the 3 cm range from the nurbs one (fig. 7). This very particular form, even though it explains the building process in a very satisfactory manner (RADOJEVIC 2015), is not admirable by the visitors of the Chapel 2. Actually, the conchoid surface (that is a solid central projection of torus), could be perceived like a transformation of torus only under certain conditions. In this paper we will give a possible way to make these characteristics evident. Design process Premise The geometrical survey based studies carried on previously by different scholars, were not successful in describing exact shape of the sail (BAGLIONI et al. 2007, SALEMI 2007). The descriptions given so far, despite drew near to geometry, did not give any explanation of the design process or of the constructive logic of such a form. The shape of the sail has been described as a form generated by series of circles that appertain to radial plans, with the variable radius. For the construction of that form infinite number of different centering would be needed. This way to construct that is not attune to Brunelleschi at all, who, just to remember, built the Dome’s cupola without fixed centering. With these premises, we will try to describe a double sailed form that is possible to build without any centering (only with one cord). ______ 2 The conchoid form, that is a three-dimensional transformation of a torus, is not perciaveble as such from the viewpoint that is different from the “transformation point”, that is the oculus. 4 Radojevic – Looking at the Pazzi Chapel’s umbrella vault from its oculus Another hypothesis, that would perfectly explain the construction issues, was the one of the toric surface; given by the two circular ribs that represent the Villarceau circles 3 (SALEMI 2007).
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